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We engineer compact contours on the moduli spaces of genus-zero Riemann surfaces that achieve analytic continuation from Euclidean to Lorentzian worldsheets. These generalized Pochhammer contours are based on the combinatorics of associahedra and make the analytic properties of tree-level amplitudes entirely manifest for any number and type of external strings. We use them in practice to perform first numerical computations of open and closed string amplitudes directly in the physical kinematics for $n=4,5,6,7,8,9$. We provide a code that allows anyone to do such computations.

We compute the worldsheet sphere partition function of string theory on global AdS$_3$ with pure NS-NS flux. Because of an unfixed Möbius symmetry on the worldsheet, there is a cancellation of infinities and only a part of the answer is unambiguous. We show that it precisely reproduces the holographic Weyl anomaly and the ambiguous terms correspond to the possible counterterms of the boundary CFT.

Abstract Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded in partial waves with all positive coefficients. While this follows (rather indirectly) from the no-ghost theorem, the simplicity of the statement and its fundamental importance for the physical consistency of string theory begs for a more direct and elementary understanding. In this note we take a step in this direction by presenting a new expression for the partial wave coefficients of string amplitudes, given by surprisingly simple double/triple contour integrals for open/closed strings. This representation allows us to directly prove unitarity of all superstring theories in D ≤ 6 spacetime dimensions, and can also be used to determine various asymptotics of the partial waves at large mass levels.

We evaluate one-loop open-string amplitudes at finite $\alpha'$ for the first time. Our method involves a deformation of the integration contour over the modular parameter $\tau$ to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop four-point amplitudes in type-I superstring theory, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants $s$ and $t$ and directly verify long-standing conjectures about their behaviour at high energies.

Abstract We propose a closed-form formula for genus 0 four-point functions in AdS3 string theory with pure NS-NS flux including arbitrary amounts of spectral flow. Our formula passes many non-trivial consistency checks and has intriguing connections to Hurwitz theory. This paper is the second in a series with several instalments.

Abstract We revisit the computation of string worldsheet correlators on Euclidean AdS3 with pure NS-NS background. We compute correlation functions with insertions of spectrally flowed operators. We explicitly solve all the known constraints of the model and for the first time conjecture a closed formula for three-point functions with arbitrary amount of spectral flow. We explain the relation of our results with previous computations in the literature and derive the fusion rules of the model. This paper is the first in a series with several installments.

Abstract We investigate the disk partition function for the open string. This is a subtle problem because of the presence of a residual gauge group PSL(2, ℝ) on the worldsheet even after fixing the conformal gauge. It naively has infinite volume and leads to a vanishing answer. We use different methods that all demonstrate that PSL(2, ℝ) effectively behaves like a group with finite negative volume in the path integral, which leads to a simple prescription for the computation of the disk partition function. We apply our findings to give a simple rederivation of the D-brane tensions.

Abstract We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. We discuss this problem with the help of the tensionless string on ℳ 3 × S 3 × T 4 $$ {\mathrm{\mathcal{M}}}_3\times {\mathrm{S}}^3\times {\mathbbm{T}}^4 $$ (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold Sym N ( T 4 $$ {\mathbbm{T}}^4 $$ ). We strengthen the analysis of [1] and show that the perturbative string partition function around a fixed bulk background already includes a sum over semi-classical geometries and large stringy corrections can be interpreted as various semi-classical geometries. We argue in particular that the string partition function on a Euclidean wormhole geometry factorizes completely into factors associated to the two boundaries of spacetime. Central to this is the remarkable property of the moduli space integral of string theory to localize on covering spaces of the conformal boundary of ℳ3. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories ⊕ N Sym N ( T 4 $$ {\mathbbm{T}}^4 $$ ). The boundary partition function is naturally expressed as a sum over winding worldsheets, each of which we interpret as a ‘stringy geometry’. We argue that the semi-classical bulk geometry can be understood as a condensate of such stringy geometries. We also briefly discuss the effect of ensemble averaging over the Narain moduli space of T 4 $$ {\mathbbm{T}}^4 $$ and of deforming away from the orbifold by the marginal deformation.

Abstract We consider string theory on AdS3 × S3 × 𝕋4 in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal AdS3, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on thermal AdS3 and the conical defect with a torus boundary is shown to agree, thus giving evidence for the equivalence of the tensionless string on these different background geometries. We also find that thermal AdS3 and the BTZ black hole are dual descriptions and the vacuum of the BTZ black hole is mapped to a single long string winding many times asymptotically around thermal AdS3. Thus the system yields a concrete example of the string-black hole transition. Consequently, reproducing the boundary partition function does not require a sum over bulk geometries, but rather agrees with the string partition function on any bulk geometry with the appropriate boundary. We argue that the same mechanism can lead to a resolution of the factorization problem when geometries with disconnected boundaries are considered, since the connected and disconnected geometries give the same contribution and we do not have to include them separately.

We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten's $i\varepsilon$ prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in $\alpha'$. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from Cutkosky rules in field theory, but bypasses the need for performing laborious sums over the intermediate states. We use this representation to study various physical aspects of string amplitudes, including their behavior in the $(s,t)$ plane, exponential suppression, decay widths of massive strings, total cross section, and low-energy expansions. We find that planar annulus amplitudes exhibit a version of low-spin dominance: at any finite energy, only a finite number of low partial-wave spins give an appreciable contribution to the imaginary part.

Abstract We introduce a bosonic ambitwistor string theory in AdS space. Even though the theory is anomalous at the quantum level, one can nevertheless use it in the classical limit to derive a novel formula for correlation functions of boundary CFT operators in arbitrary space-time dimensions. The resulting construction can be treated as a natural extension of the CHY formalism for the flat-space S-matrix, as it similarly expresses tree-level amplitudes in AdS as integrals over the moduli space of Riemann spheres with punctures. These integrals localize on an operator-valued version of scattering equations, which we derive directly from the ambitwistor string action on a coset manifold. As a testing ground for this formalism we focus on the simplest case of ambitwistor string coupled to two cur- rent algebras, which gives bi-adjoint scalar correlators in AdS. In order to evaluate them directly, we make use of a series of contour deformations on the moduli space of punctured Riemann spheres and check that the result agrees with tree level Witten diagram computations to all multiplicity. We also initiate the study of eigenfunctions of scattering equations in AdS, which interpolate between conformal partial waves in different OPE channels, and point out a connection to an elliptic deformation of the Calogero-Sutherland model.

Abstract We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT u $$ \mathfrak{u} $$ (M + N ) k /( u $$ \mathfrak{u} $$ (M ) k × u $$ \mathfrak{u} $$ (N ) k ). This VOA serves as a basic building block for a large class of cosets and generalizes the W ∞ $$ {\mathcal{W}}_{\infty } $$ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the N $$ \mathcal{N} $$ = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.

Abstract We continue our study of the worldsheet theory of superstrings on AdS3 × S3 × 𝕋4 in the tensionless limit [1]. We consider the theory on higher genus surfaces. We give evidence that the worldsheet correlators localise on certain worldsheets that cover the boundary of AdS3 holomorphically. This simplifies the string moduli space integral dramatically to a finite sum. This property shows that the higher genus corrections of the string worldsheet reproduce the structure of the 1/N corrections in the dual symmetric orbifold CFT Sym N (𝕋4).

Abstract It was recently argued that string theory on AdS3 × S3 × 𝕋4 with one unit (k = 1) of NS-NS flux is exactly dual to the symmetric orbifold CFT Sym N (𝕋4). In this paper we show how to directly relate the n-point correlators of the two sides to one another. In particular, we argue that the correlators of the world-sheet theory are delta-function- localised in string moduli space to those configurations that allow for a holomorphic covering map of the S2-boundary of AdS3 by the world-sheet. This striking feature can be seen both from a careful Ward identity analysis, as well as from semi-classically exact AdS3 solutions that are pinned to the boundary. The world-sheet correlators therefore have exactly the same structure as in the Lunin-Mathur construction of symmetric orbifold CFT correlators in terms of a covering surface — which now gets identified with the world-sheet. Together with the results of [1, 2] this essentially demonstrates how the k = 1 AdS3 string theory becomes equivalent to the spacetime orbifold CFT in the genus expansion.

Abstract We exploit null vectors of the fractional Virasoro algebra of the symmetric product orbifold to compute correlation functions of twist fields in the large N limit. This yields a new method to derive correlation functions in these orbifold CFTs that is purely based on the symmetry algebra. We explore various generalisations, such as subleading (torus) contributions or correlation functions of other fields than the bare twist fields. We comment on the consequences of our computation for the AdS3 /CFT2 correspondence.

Abstract Recently, string theory on AdS3 × S3 × 𝕋4 with one unit of NS-NS flux (k = 1) was argued to be exactly dual to the symmetric orbifold of 𝕋 4, and in particular, the full (unprotected) spectrum was matched between the two descriptions. This duality was later extended to the case with higher NS-NS background flux for which the long string sector was argued to be described by the symmetric product orbifold of ( N $$ \mathcal{N} $$ = 4 Liouville) × 𝕋 4. In this paper we study correlation functions for the bosonic analogue of this duality, relating bosonic string theory on AdS3 × X to the symmetric orbifold of Liouville × X. More specifically, we show that the low-lying null vectors of Liouville theory correspond to BRST exact states from the worldsheet perspective, and we demonstrate that they give rise to the expected BPZ differential equations for the dual CFT correlators. Since the structure constants of Liouville theory are uniquely fixed by these constraints, this shows that the seed theory of the dual CFT contains indeed the Liouville factor.

Abstract We construct a quadratic basis of generators of matrix-extended W 1 + ∞ $$ {\mathcal{W}}_{1+\infty } $$ using a generalization of the Miura transformation. This makes it possible to conjecture a closed-form formula for the operator product expansions defining the algebra. We study truncations of the algebra. An explicit calculation at low levels shows that these are parametrized in a way consistent with the gluing description of the algebra. It is perhaps surprising that in spite of the fact that the algebras are rather complicated and non-linear, the structure of their truncations follows very simple gluing rules.

Abstract String theory on AdS3 × S3 × S3 × S1 with pure NS-NS flux and minimal flux through one of the two S3’s is studied from a world-sheet perspective. It is shown that the spacetime spectrum, as well as the algebra of spectrum generating operators, matches precisely that of the symmetric orbifold of S3 × S1 in the large N limit. This gives strong support for the proposal that these two descriptions are exactly dual to one another.

Abstract Superstring theory on AdS 3 × S 3 × T 4 $$ {\mathrm{AdS}}_3 \times {S}^3\times {\mathbb{T}}^4 $$ with the smallest amount of NS-NS flux (“k = 1”) is shown to be dual to the spacetime CFT given by the large N limit of the free symmetric product orbifold Sym N T 4 $$ \left({\mathbb{T}}^4\right) $$ . To define the worldsheet theory at k = 1, we employ the hybrid formalism in which the AdS3 × S3 part is described by the p s u 1 , 1 | 2 1 $$ \mathfrak{p}\mathfrak{s}\mathfrak{u}{\left(1,1\Big|2\right)}_1 $$ WZW model (which is well defined). Unlike the case for k ≥ 2, it turns out that the string spectrum at k = 1 does not exhibit the long string continuum, and perfectly matches with the large N limit of the symmetric product. We also demonstrate that the fusion rules of the symmetric orbifold are reproduced from the worldsheet perspective. Our proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory, for which the dual CFT is also explicitly known.

Abstract We revisit two related phenomena in AdS3 string theory backgrounds. At pure NS-NS flux, the spectrum contains a continuum of long strings which can escape to the boundary of AdS3 at a finite cost of energy. Related to this are certain gaps in the BPS spectrum one computes from the RNS worldsheet description. One expects that both these effects disappear when perturbing slightly away from the pure NS-NS flux background. We employ the hybrid formalism for mixed flux backgrounds to demonstrate directly from the worldsheet that this is indeed the case.

Abstract We study string theory on AdS3 backgrounds with mixed flux using the hybrid formalism of Berkovits, Vafa and Witten. We solve the worldsheet description of the theory completely in the plane-wave limit. This constitutes a direct derivation of the plane-wave spectrum from the worldsheet with mixed flux.

For string theory on AdS3 with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows us to show that the long string sector of superstring theory on AdS3×S3×T4 for generic NS-NS flux is described by the symmetric orbifold of (N=4Liouville theory)×T4. For the case of minimal flux (k=1), the Liouville factor disappears, and we just obtain the symmetric orbifold of T4, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on AdS×3X.

Abstract We consider string theory on AdS3 × (S3 × S3 × S1)/ℤ2, a background supporting N=33 $$ \mathcal{N}=\left(3,3\right) $$ spacetime supersymmetry. We propose that string theory on this background is dual to the symmetric product orbifold of S0/ℤ2 $$ {\mathcal{S}}_0/{\mathbb{Z}}_2 $$ where S0 $$ {\mathcal{S}}_0 $$ is a theory of four free fermions and one free boson. We show that the BPS spectra of the two sides of the duality match precisely. Furthermore, we compute the elliptic genus of the dual CFT and that of the supergravity limit of string theory and demonstrate that they match, hence providing non-trivial support for the holographic proposal.

Abstract A new family of higher spin algebras that arises upon restricting matrix extensions of s h s λ $$ \mathfrak{s}\mathfrak{h}\mathfrak{s}\left[\lambda \right] $$ is found. We identify coset CFTs realising these symmetry algebras, and thus propose new higher spin-CFT dual pairs. These higher spin theories arise naturally as a subsector of string theory on AdS3 × S3 × S3 × S1 for specific ratios of the radii of the two spheres.

Abstract We analyse the conditions for AdS3 × ℳ7 backgrounds with pure NS-NS flux to be supersymmetric. We classify all N=2,2 $$ \mathcal{N}=\left(2,\ 2\right) $$ solutions where ℳ7 satisfies the stronger condition of being a U(1)-fibration over a Kähler manifold. We compute the BPS spectrum of all the backgrounds in this classification. We assign a natural dual CFT to the backgrounds and confirm that the BPS spectra agree, thus providing evidence in favour of the proposal.

Abstract We propose a class of AdS3/CFT2 dualities with N $$ \mathcal{N} $$ = (2, 2) supersymmetry. These dualities relate string theory on AdS3×S3×T4/G $$ {\mathrm{AdS}}_3\times \left({\mathrm{S}}^3\times {\mathbb{T}}^4\right)/\mathrm{G} $$ to marginal deformations of the symmetric product orbifold of T4/G $$ {\mathbb{T}}^4/\mathrm{G} $$, where G is a dihedral group. We demonstrate that the BPS spectrum calculated from supergravity and string theory agrees with that of the dual CFT. Moreover, the supergravity elliptic genus is shown to reproduce the CFT answer, thus providing further non-trivial evidence in favour of the proposal.

Abstract The CFT dual of string theory on AdS3 × S3 × S3 × S1 is conjectured to be the symmetric orbifold of the S κ $$ {\mathcal{S}}_{\kappa } $$ theory, provided that one of the two Q 5 ± quantum numbers is a multiple of the other. We determine the BPS spectrum of the symmetric orbifold in detail, and show that it reproduces precisely the BPS spectrum that was recently calculated in supergravity. We also determine the BPS spectrum of the world-sheet theory that is formulated in terms of WZW models, and show that, apart from some gaps (which are reminiscent of those that appear in the corresponding T 4 $$ {\mathbb{T}}^4 $$ calculation), it also reproduces this BPS spectrum. In fact, the matching seems to work as well as for the familiar T 4 $$ {\mathbb{T}}^4 $$ case, and thus our results give strong support for this proposal.